ORBi Collection: Mathematics
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How to extract the oscillating components of a signal? A wavelet-based approach compared to the Empirical Mode Decomposition
http://hdl.handle.net/2268/205402
Title: How to extract the oscillating components of a signal? A wavelet-based approach compared to the Empirical Mode Decomposition
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<br/>Author, co-author: Deliège, AdrienMon, 16 Jan 2017 17:13:32 GMTMars Topography Investigated Through the Wavelet Leaders Method: a Multidimensional Study of its Fractal Structure
http://hdl.handle.net/2268/205397
Title: Mars Topography Investigated Through the Wavelet Leaders Method: a Multidimensional Study of its Fractal Structure
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<br/>Author, co-author: Deliège, Adrien; Kleyntssens, Thomas; Nicolay, Samuel
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<br/>Abstract: This work examines the scaling properties of Mars topography through a wavelet-based formalism. We conduct exhaustive one-dimensional (both longitudinal and latitudinal) and two-dimensional studies based on Mars Orbiter Laser Altimeter (MOLA) data using the multifractal formalism called Wavelet Leaders Method (WLM). This approach shows that a scale break occurs at approximately 15 km, giving two scaling regimes in both 1D and 2D cases. At small scales, these topographic profiles mostly display a monofractal behavior while a switch to multifractality is observed in several areas at larger scales. The scaling exponents extracted from this framework tend to be greater at small scales. In the 1D context, these observations are in agreement with previous works and thus suggest that the WLM is well-suited for examining scaling properties of topographic fields. Moreover, the 2D analysis is the first such complete study to our knowledge. It gives both a local and global insight on the scaling regimes of the surface of Mars and allows to exhibit the link between the scaling exponents and several famous features of the Martian topography. These results may be used as a solid basis for further investigations of the scaling laws of the Red planet and show that the WLM could be used to perform systematic analyses of the surface roughness of other celestial bodies.Mon, 16 Jan 2017 16:56:12 GMTIs Büchi's theorem useful for you? (for an audience of logicians)
http://hdl.handle.net/2268/205384
Title: Is Büchi's theorem useful for you? (for an audience of logicians)
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<br/>Author, co-author: Rigo, Michel
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<br/>Abstract: Almost a century ago, Presburger showed that the first order theory of the natural numbers with addition is decidable. Following the work of B\"uchi in 1960, this result still holds when adding a function $V_k$ to the structure, where $V_k(n)$ is the largest power of $k\ge 2$ diving $n$. In particular, this leads to a logical characterization of the $k$-automatic sequences.
During the last few years, many applications of this result have been considered in combinatorics on words, mostly by J. Shallit and his coauthors.
In this talk, we will present this theorem of B\"uchi where decidability relies on finite automata. Then we will review some results about automatic sequences or morphic words that can be proved automatically (i.e., the proof is carried on by an algorithm). Finally, we will sketch the limitation of this technique. With a single line formula, one can prove automatically that the Thue-Morse word has no overlap but, hopefully, not all the combinatorial properties of morphic words can be derived in this way.
We will not assume any background in combinatorics on words from the audience.Mon, 16 Jan 2017 15:11:16 GMTSurprenante beauté des mathématiques
http://hdl.handle.net/2268/205338
Title: Surprenante beauté des mathématiques
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<br/>Author, co-author: Bair, Jacques
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<br/>Abstract: Dans cet article, nous dissertons sur le côté esthétique des mathématiques. Selon nous, la beauté des mathématiques apparaît souvent au travers de raisonnements lumineux mais surprenants. Nous illustrons ce point au moyen de trois exemples concrets, simples et empruntés au mathématicien russe Lazebnik.Mon, 16 Jan 2017 10:14:42 GMTSummatory function of sequences counting subwords occurrences
http://hdl.handle.net/2268/205328
Title: Summatory function of sequences counting subwords occurrences
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<br/>Author, co-author: Leroy, Julien; Rigo, Michel; Stipulanti, ManonMon, 16 Jan 2017 09:04:05 GMTQuelques idées générales à propos de la compréhension en mathématiques
http://hdl.handle.net/2268/205282
Title: Quelques idées générales à propos de la compréhension en mathématiques
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<br/>Author, co-author: Bair, Jacques
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<br/>Abstract: Dans cet article, nous apportons des éléments de réponse à la question générale suivante : "Que signifie bien comprendre en mathématiques ?" Nous nous demandons pourquoi il est difficile d'y répondre et apportons une piste de solution en distinguant les activités de production de celles de reproduction. De plus, nous abordons le problème étudié d'un point de vue aussi bien local que global. Enfin, nous nous interrogeons sur l'incompréhension en mathématiques en proposant une typologie, en repérant des causes possibles d'incompréhension et en donnant quelques conseils pratiques.Fri, 13 Jan 2017 14:48:58 GMTLe concept de duration : une présentation heuristique
http://hdl.handle.net/2268/205225
Title: Le concept de duration : une présentation heuristique
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<br/>Author, co-author: Bair, Jacques
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<br/>Abstract: La duration est fondamentale en mathématiques ﬁnancières. Nous la présentons de deux manières complémentaires qui sont qualiﬁées de classique et d’heuristique. La seconde méthode nous paraît originale et bien adaptée pour amener les étudiants à eﬀectuer un travail de recherche sur un problème concret et pour appliquer de nombreux concepts classiquement enseignés dans les programmes de mathématiques.Thu, 12 Jan 2017 14:38:06 GMTPensées (mathématiques) de Tao
http://hdl.handle.net/2268/205141
Title: Pensées (mathématiques) de Tao
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<br/>Author, co-author: Bair, Jacques
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<br/>Abstract: Après une biographie de Terence Tao, certaines de ses idées sont développées à propos de ce que sont les bonnes mathématiques, des étapes dans l'apprentissage des mathématiques et de quelques conseils pratiques donnés à l'intention d'apprenants au niveau de la transition entre le secondaire et le supérieur.Wed, 11 Jan 2017 12:05:54 GMTThe constant of recognizability is computable for primitive morphisms
http://hdl.handle.net/2268/205116
Title: The constant of recognizability is computable for primitive morphisms
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<br/>Author, co-author: Durand, Fabien; Leroy, JulienTue, 10 Jan 2017 14:24:00 GMTDecidable properties of extension graphs for substitutive languages
http://hdl.handle.net/2268/205115
Title: Decidable properties of extension graphs for substitutive languages
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<br/>Author, co-author: Dolce, Francesco; Leroy, Julien; Kyriakoglou, RevekkaTue, 10 Jan 2017 14:20:52 GMTGeneralized Pascal triangles for binomial coefficients of words: a short introduction
http://hdl.handle.net/2268/205079
Title: Generalized Pascal triangles for binomial coefficients of words: a short introduction
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<br/>Author, co-author: Stipulanti, Manon
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<br/>Abstract: We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a finite word appears as a subsequence of another finite word. Similarly to the Sierpiński gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo 2, we describe and study the first properties of the subset of [0, 1] × [0, 1] associated with this extended Pascal triangle modulo a prime p.
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<br/>Commentary: Work in collaboration with Julien Leroy (ULg, j.leroy@ulg.ac.be) and Michel Rigo (ULg, m.rigo@ulg.ac.be). // Travail en collaboration avec Julien Leroy (ULg, j.leroy@ulg.ac.be) et Michel Rigo (ULg, m.rigo@ulg.ac.be).Mon, 09 Jan 2017 15:56:14 GMTCounting the number of non-zero coefficients in rows of generalized Pascal triangles
http://hdl.handle.net/2268/205077
Title: Counting the number of non-zero coefficients in rows of generalized Pascal triangles
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<br/>Author, co-author: Leroy, Julien; Rigo, Michel; Stipulanti, Manon
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<br/>Abstract: This paper is about counting the number of distinct (scattered) subwords occurring in a given word. More precisely, we consider the generalization of the Pascal triangle to binomial coefficients of words and the sequence (S(n))n≥0 counting the number of positive entries on each row. By introducing a convenient tree structure, we provide a recurrence relation for (S(n))n≥0. This leads to a connection with the 2-regular Stern–Brocot sequence and the sequence of denominators occurring in the Farey tree. Then we extend our construction to the Zeckendorf numeration system based on the Fibonacci sequence. Again our tree structure permits us to obtain recurrence relations for and the F-regularity of the corresponding sequence.Mon, 09 Jan 2017 14:34:37 GMTBispecial Factors in the Brun S-Adic System
http://hdl.handle.net/2268/205075
Title: Bispecial Factors in the Brun S-Adic System
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<br/>Author, co-author: Labbé, Sébastien; Leroy, JulienMon, 09 Jan 2017 13:52:01 GMTInterpreting the Infinitesimal Mathematics of Leibniz and Euler
http://hdl.handle.net/2268/204854
Title: Interpreting the Infinitesimal Mathematics of Leibniz and Euler
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<br/>Author, co-author: Bair, Jacques; Blaszczyk, Piotr; Ely, Robert; Henry, Valérie; Kanovei, Vladimir; Katz, Karin U.; Katz, Mikhail G.; Kutateladze, Semen S.; McGaffey, Thomas; Reeder, Patrick; Schaps, David M.; Sherry, David; Schnider, Steven
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<br/>Abstract: We apply Benacerraf’s distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice) to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves.Tue, 03 Jan 2017 13:28:31 GMTInterpreting the Infinitesimal Mathematics of Leibniz and Euler
http://hdl.handle.net/2268/204854
Title: Interpreting the Infinitesimal Mathematics of Leibniz and Euler
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<br/>Author, co-author: Bair, Jacques; Blaszczyk, Piotr; Ely, Robert; Henry, Valérie; Kanovei, Vladimir; Katz, Karin U.; Katz, Mikhail G.; Kutateladze, Semen S.; McGaffey, Thomas; Reeder, Patrick; Schaps, David M.; Sherry, David; Schnider, Steven
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<br/>Abstract: We apply Benacerraf’s distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice) to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves.Tue, 03 Jan 2017 13:28:31 GMTInterpreting the Infinitesimal Mathematics of Leibniz and Euler
http://hdl.handle.net/2268/204854
Title: Interpreting the Infinitesimal Mathematics of Leibniz and Euler
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<br/>Author, co-author: Bair, Jacques; Blaszczyk, Piotr; Ely, Robert; Henry, Valérie; Kanovei, Vladimir; Katz, Karin U.; Katz, Mikhail G.; Kutateladze, Semen S.; McGaffey, Thomas; Reeder, Patrick; Schaps, David M.; Sherry, David; Schnider, Steven
<br/>
<br/>Abstract: We apply Benacerraf’s distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice) to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves.Tue, 03 Jan 2017 13:28:31 GMTChapter VIII "Equations and languages" in J.-É. Pin, Mathematical Foundations of Automata Theory
http://hdl.handle.net/2268/204751
Title: Chapter VIII "Equations and languages" in J.-É. Pin, Mathematical Foundations of Automata Theory
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<br/>Author, co-author: Stipulanti, Manon
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<br/>Abstract: We present the chapter VIII titled "Equations and Langages" in Jean-Éric Pin, Mathematical Foundations of Automata Theory.Tue, 27 Dec 2016 14:58:36 GMTChapter 2 "Substitutions, arithmetic and finite automata: an introduction" in Pytheas Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics
http://hdl.handle.net/2268/204750
Title: Chapter 2 "Substitutions, arithmetic and finite automata: an introduction" in Pytheas Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics
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<br/>Author, co-author: Stipulanti, Manon
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<br/>Abstract: This presentation is about the chapter 2 titled "Substitutions, arithmetic and finite automata: an introduction" in Pytheas Fogg N., Substitutions in dynamics, arithmetics, and combinatorics. Berlin : Springer, 2002.Tue, 27 Dec 2016 14:47:50 GMTMATH2010-1 Logiciels mathématiques - Notes de cours
http://hdl.handle.net/2268/204748
Title: MATH2010-1 Logiciels mathématiques - Notes de cours
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<br/>Author, co-author: Labbé, SébastienTue, 27 Dec 2016 14:35:23 GMTSébastien Labbé Research Code: slabbe 0.3.b2
http://hdl.handle.net/2268/204746
Title: Sébastien Labbé Research Code: slabbe 0.3.b2
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<br/>Author, co-author: Labbé, SébastienTue, 27 Dec 2016 14:16:01 GMT